Embeddings of manifolds with boundary: classification

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For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$3]{Skopenkov2016c}.
For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$3]{Skopenkov2016c}.
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If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
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<!--If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.-->
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We state the results below in piecewise-linear (PL) category.
We do not provide references to the original proofs of stated results.
We do not provide references to the original proofs of stated results.

Revision as of 12:57, 26 March 2020


This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

Recall that some Unknotting Theorems hold for manifolds with boundary [Skopenkov2016c, \S3], [Skopenkov2006, \S2]. In this page we present results peculiar for manifold with non-empty boundary.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3].

We state the results below in piecewise-linear (PL) category.

We do not provide references to the original proofs of stated results.

Theorem 1.1. Every n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

This result can be found in [Horvatic1971, theorem 5.2]


2 Unknotting Theorems

Theorem 2.1. Assume N is a compact connected n-manifold and either m \ge 2n+2 or m\ge2n+1\ge5. Then any two embeddings of N into \R^m are isotopic.

The condition m \ge 2n+2 stands for General Position Theorem and the condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see theorems 2.1 and 2.2 respectivly of [Skopenkov2016c, \S 2].

Theorem 2.2. Assume that N is a compact connected n-manifold with non-empty boundary and one of the following conditions holds:

(a) m \ge 2n

(b) N is 1-connected, m \ge 2n - 1\ge3

Then any two embeddings of N into \R^m are isotopic.

Part (a) of this theorem in case n>2 can be found in [Edwards1968, \S 4, corollary 5]. Case n=1 is clear. Case n=2 can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, \S 5]. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see [Penrose&Whitehead&Zeeman1961, Theorem 1.2b].

Theorem 2.3. [The Haefliger-Zeeman unknotting Theorem] For every n\ge2k + 2, m \ge 2n - k + 1 and closed k-connected n-manifold N, any two embeddings of N into \R^m are isotopic.

Theorem 2.4. For every m\ge2n-k and m-k\ge3 and k-connected n-manifold N with non-empty boundary, any two embeddings of N into \R^m are isotopic.

This result can be found in [Hudson1969, Theorem 10.3]


3 Construction and examples

...

4 Invariants

...

5 Classification

Theorem 5.1.[Becker-Glover] Let N be a closed homologically k-connected n-manifold and m\ge 3n/2+2. The cone map \Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N) is one-to-one for m\ge 2n-2k and is surjective for m=2n-2k-1.

[Becker&Glover1971]

6 Further discussion

...

7 References

, $\S]{Skopenkov2016c}. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. We do not provide references to the original proofs of stated results. {{beginthm|Theorem}} Every $n$-manifold $N$ with nonempty boundary PL embeds into $\R^{2n-1}$. {{endthm}} This result can be found in \cite[theorem 5.2]{Horvatic1971} == Unknotting Theorems == ; {{beginthm|Theorem}}\label{th::unknotting} Assume $N$ is a compact connected $n$-manifold and either $m \ge 2n+2$ or $m\ge2n+1\ge5$. Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} The condition $m \ge 2n+2$ stands for General Position Theorem and the condition $m\ge2n+1\ge5$ stands for Whitney-Wu Unknotting Theorem, see [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|theorems 2.1 and 2.2]] respectivly of \cite[$\S$ 2]{Skopenkov2016c}. {{beginthm|Theorem}} Assume that $N$ is a compact connected $n$-manifold with non-empty boundary and one of the following conditions holds: (a) $m \ge 2n$ (b) $N$ is \S3], [Skopenkov2006, \S2]. In this page we present results peculiar for manifold with non-empty boundary.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3].

We state the results below in piecewise-linear (PL) category.

We do not provide references to the original proofs of stated results.

Theorem 1.1. Every n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

This result can be found in [Horvatic1971, theorem 5.2]


2 Unknotting Theorems

Theorem 2.1. Assume N is a compact connected n-manifold and either m \ge 2n+2 or m\ge2n+1\ge5. Then any two embeddings of N into \R^m are isotopic.

The condition m \ge 2n+2 stands for General Position Theorem and the condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see theorems 2.1 and 2.2 respectivly of [Skopenkov2016c, \S 2].

Theorem 2.2. Assume that N is a compact connected n-manifold with non-empty boundary and one of the following conditions holds:

(a) m \ge 2n

(b) N is 1-connected, m \ge 2n - 1\ge3

Then any two embeddings of N into \R^m are isotopic.

Part (a) of this theorem in case n>2 can be found in [Edwards1968, \S 4, corollary 5]. Case n=1 is clear. Case n=2 can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, \S 5]. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see [Penrose&Whitehead&Zeeman1961, Theorem 1.2b].

Theorem 2.3. [The Haefliger-Zeeman unknotting Theorem] For every n\ge2k + 2, m \ge 2n - k + 1 and closed k-connected n-manifold N, any two embeddings of N into \R^m are isotopic.

Theorem 2.4. For every m\ge2n-k and m-k\ge3 and k-connected n-manifold N with non-empty boundary, any two embeddings of N into \R^m are isotopic.

This result can be found in [Hudson1969, Theorem 10.3]


3 Construction and examples

...

4 Invariants

...

5 Classification

Theorem 5.1.[Becker-Glover] Let N be a closed homologically k-connected n-manifold and m\ge 3n/2+2. The cone map \Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N) is one-to-one for m\ge 2n-2k and is surjective for m=2n-2k-1.

[Becker&Glover1971]

6 Further discussion

...

7 References

$-connected, $m \ge 2n - 1\ge3$ Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} Part (a) of this theorem in case $n>2$ can be found in \cite[$\S$ 4, corollary 5]{Edwards1968}. Case $n=1$ is clear. Case $n=2$ can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion \cite[$\S$ 5]{Skopenkov2006}. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see \cite[Theorem 1.2b]{Penrose&Whitehead&Zeeman1961}. {{beginthm|Theorem}} [The Haefliger-Zeeman unknotting Theorem] For every $n\ge2k + 2$, $m \ge 2n - k + 1$ and closed $k$-connected $n$-manifold $N$, any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} {{beginthm|Theorem}} For every $m\ge2n-k$ and $m-k\ge3$ and $k$-connected $n$-manifold $N$ with non-empty boundary, any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} This result can be found in \cite[Theorem 10.3]{Hudson1969} == Construction and examples == ; ... == Invariants == ; ... == Classification == ; {{beginthm|Theorem}}[Becker-Glover] Let $N$ be a closed homologically $k$-connected $n$-manifold and $m\ge 3n/2+2$. The cone map $\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$ is one-to-one for $m\ge 2n-2k$ and is surjective for $m=2n-2k-1$. {{endthm}} \cite{Becker&Glover1971} == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S3], [Skopenkov2006, \S2]. In this page we present results peculiar for manifold with non-empty boundary.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3].

We state the results below in piecewise-linear (PL) category.

We do not provide references to the original proofs of stated results.

Theorem 1.1. Every n-manifold N with nonempty boundary PL embeds into \R^{2n-1}.

This result can be found in [Horvatic1971, theorem 5.2]


2 Unknotting Theorems

Theorem 2.1. Assume N is a compact connected n-manifold and either m \ge 2n+2 or m\ge2n+1\ge5. Then any two embeddings of N into \R^m are isotopic.

The condition m \ge 2n+2 stands for General Position Theorem and the condition m\ge2n+1\ge5 stands for Whitney-Wu Unknotting Theorem, see theorems 2.1 and 2.2 respectivly of [Skopenkov2016c, \S 2].

Theorem 2.2. Assume that N is a compact connected n-manifold with non-empty boundary and one of the following conditions holds:

(a) m \ge 2n

(b) N is 1-connected, m \ge 2n - 1\ge3

Then any two embeddings of N into \R^m are isotopic.

Part (a) of this theorem in case n>2 can be found in [Edwards1968, \S 4, corollary 5]. Case n=1 is clear. Case n=2 can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006, \S 5]. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see [Penrose&Whitehead&Zeeman1961, Theorem 1.2b].

Theorem 2.3. [The Haefliger-Zeeman unknotting Theorem] For every n\ge2k + 2, m \ge 2n - k + 1 and closed k-connected n-manifold N, any two embeddings of N into \R^m are isotopic.

Theorem 2.4. For every m\ge2n-k and m-k\ge3 and k-connected n-manifold N with non-empty boundary, any two embeddings of N into \R^m are isotopic.

This result can be found in [Hudson1969, Theorem 10.3]


3 Construction and examples

...

4 Invariants

...

5 Classification

Theorem 5.1.[Becker-Glover] Let N be a closed homologically k-connected n-manifold and m\ge 3n/2+2. The cone map \Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N) is one-to-one for m\ge 2n-2k and is surjective for m=2n-2k-1.

[Becker&Glover1971]

6 Further discussion

...

7 References

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